RADEMACHER CHAOS IN SYMMETRIC SPACES, II

S. V. ASTASHKINDepartment of Mathematics, Samara State University,ul. Akad. Pavlova 1, 443011 Samara, RUSSIAE-mail: astashkn@ssu.samara.ruAbstract. In this paper we study some properties of the orthonormal system {rirj}1i<j<where rk(t) are Rademacher functions on [0, 1], k = 1, 2, . . . This system is usually called Rademacherchaos of order 2. It is shown that a specic ordering of the chaos leads to a basic sequence (pos-sibly non-unconditional) in a wide class of symmetric functional spaces on [0, 1]. Necessary andsucient conditions on the space are found for the basic sequence {rirj}1i<j< to possess theunconditionality property.1. IntroductionThis paper is a continuation of [1] where we started the study of Rademacher chaos in functionalsymmetric spaces (s.s.) on the segment [0, 1]. Let us rst recall some denitions and notations from[1].As usual,rk(t) = sign sin 2k1t (k = 1, 2, . . .)denotes the system of Rademacher functions on I := [0, 1]. The set of all real-valued functions x(t)that can be represented in the formx(t) =

1i<j<ai,jri(t)rj(t) (t [0, 1])is called a chaos of order 2 with respect to the system rk(t) (Rademacher chaos of order 2 ). Thesame name is used, with no ambiguity, for the orthonormal system of functions rirj

1i<j<. Inthe sequel, as in [1], H denotes the closure of L

in the Orlicz space L

M where M(t) = et1.In [1], we proved the following.Theorem A. Let X be a symmetric space. Then the following statements are equivalent:1) The system rirj

1i<j< in X is equivalent to the canonical basis in l2;2) A continuous imbedding H X takes place.In this paper, we shall consider questions related to the unconditionality of Rademacher chaos.Our main result is:The statements 1) and 2) in Theorem A are equivalent to the next one:3) The system rirj

1i<j< is an unconditional basic sequence in s.s. X.Let us recall the meaning of the central notions above.Denition. A sequence xn

n=1 of elements in Banach space X is called a basic sequence if it isa basis in its closed linear span [xn]

n=1.As is well-known (see for example [11, p.2] ), the latter is equivalent to the following two conditions:1) xn ,= 0 for all n N;2) The family of projectorsPm_

i=1aixi_ =m

i=1aixi (m = 1, 2, . . .),1dened on [xn]

n=1, is uniformly bounded. That is, a constant K > 0 exists such that for all m, n N,m < n, and ai R, the following inequality holds:(1)____m

i=1aixi____ K____n

i=1aixi____.One of the most important properties of a basic sequence is its unconditionality.Denition. A basic sequence xn

n=1 in a Banach space X is said to be unconditional if, for anyrearrangement of N, the sequence x(n)

n=1 is also a basic sequence in X.This is equivalent, in particular, to the uniform boundedness of the family of operatorsM

i=1aixi_ =

i=1

iaixi (i = 1)which are dened on [xn]

n=1 [11, p.18], and the last means that there is a constant K0 such that foreach n N and any couple of sequences of signs i and real numbers ai,(2)____n

i=1

iaixi____ K0____n

i=1aixi____.Finally, note that for sequences of real numbers (ai,j)1i<j< we use the common notation|(ai,j)|2 :=_

k(k1)/2+1 = r1rk+1 = rk+1, . . . , k(k+1)/2 = rkrk+1, . . .Before formulating our rst theorem let us recall the denition of a fundamental notion in theinterpolation theory of operators (for more details, see [8]).Denition. A Banach space X is said to be an interpolation space with respect to the Banachcouple (X0, X1) if X0X1 X X0+X1 and, in addition, if the boundedness of a linear operatorT in both X0 and X1 implies its boundedness in X as well.Theorem 1. The Rademacher chaos rirj

1i<j<, ordered according to rule (3), is a basicsequence in every interpolation with respect to the couple (L1, L

) |z|X |ky|X + |f|X 3B|y|X.The denitions of the functions z and y, together with inequalities (6) and (6), yield that relation(1) holds true for the Rademacher chaos which is ordered according to (3). The theorem is proved.Remark 1. The requirement for the space X to be an interpolation space with respect to thecouple (L1, L

) is not very restrictive. The most important s.s. (Orlicz, Lorentz, Marcinkiewiczspaces and others) possess this property [8, p.142]. In addition, it is seen from the proof of thetheorem that the above condition may be replaced by a weaker one: the boundedness in X of theaveraging operators corresponding to the dyadic partitionings of the interval [0, 1].33. Rademacher chaos as unconditional basic sequenceWe go now further to the study of the unconditionality of Rademacher chaos in s.s. We have alreadymentioned that the main result in this paper amplies Theorem A proved in [1] and formulated inSection 1.Theorem 2. Let X be s.s. on [0, 1]. Then the following assertions are equivalent:1) The system rirj

1i<j< in X is equivalent to the canonical basis in the space l2, that is,there is a constant C > 0 that depends only on the space X; such that for all real numbers ai,j (1 i < j < ),(7) C1|(ai,j)|2

1i<j< is an unconditional basic sequence in X.Remark 2. The implication 1) 3) is evident and the equivalence 1) 2) is proved in [1]. Thus,it suces to prove the implication 3) 1).First, we prove a weaker assertion. Let G denote the closure of L

in the Orlicz space L

Ncorresponding to the function N(t) = et21.Proposition 1. Let the s.s. X on [0, 1] be such that X G and the system rirj

1i<j< in X is equivalent to the canonical basis in l2.For the proof we need a lemma that concerns spaces with a mixed norm. Let us recall the denition(for details, see [5, p.400] ).Denition. Let X and Y be s.s. on [0, 1]. The space with a mixed norm X[Y ] is the set of allmeasurable functions x(s, t) on the square I I satisfying the conditions:1) x(, t) Y for almost all t I;2) x(t) := |x(, t)|Y X.Dene|x|X[Y ] = |x|X.Let A = A(u) be a N-function on [0, ). This means that A is continuous, convex, and satiseslimu+0A(u)u = limu+uA(u) = 0.As usual, denote by LA the Orlicz space of all functions x = x(t) measurable on [0, 1] and having anite norm,|x|LA := inf_ > 0 :_ 10A_[x(t)[

1i<j< together with the basiccoecients is a RUC-system in X.Corollary 2. For each s.s. X the following assertions are equivalent:1) X G;2) _10____

1i<j<ai,jri,j(u)rirj____Xdu |(ai,j)|2.Proof. The implication 1) 2) follows from inequalities (8) and (10).Suppose now that 2) takes place and let ai,j = 0 (i ,= 1). From the denition of Rademacherfunctions and the assumptions we get_ 10____

1i<j<ai,jri,j(u)rirj____Xdu C|(ai,j)|2.Now the assertion follows from the fact that the opposite inequality holds always (see (10)).In order to prove Theorem 2 we need some more auxiliary assertions.Let nk

l=1clml for t E.Inequality (11) follows now from the denition of the rearrangement and the fact that [E[ = tk.The next assertion makes Theorem 8 from [1] more precise. We shall use here the same notationsas in [1].If X is a s.s. on [0, 1], then

R(X) denotes a subspace of X consisting of all functions of the formx(t) =

1i<j<, zk and yk (k = 1, 2, . . .) be dened in the same way as in theproof of Proposition 2. It is well-known [13] that Marcinkiewicz space M(1/2) = M(t log2(2/t))coincides with the Orlicz space LM, M(t) = et 1. Therefore, by Theorem A from Section 1, wehave(15) |zk|M(1/2)

) for some (1/2, 1/2). If for any

arrangement of signs = i,j

1i<j< the operator

T

is bounded in

R(X), thenX _0<<1/4/2M(

).The proof is similar to that of Corollary 4.Now we are ready to prove our main Theorem 2.Proof of Theorem 2. As it has been already mentioned in Remark 2, it suces to verify theimplication 3) 1).If the system rirj

1i<j< is unconditional in s.s. X, then, for each arrangement of signs , theoperator

T

is bounded in

R(X). In particular, we get by Corollary 4 that X M(1/5). Therefore,by Corollary 5, it follows that X M(1/10). Since M(1/10) G, then all the more, X G.Finally, applying Proposition 1 we conclude that the system rirj

1i<j< is equivalent to thecanonical basis in l2. The theorem is proved.Remark 3. Assertions analogous to Theorems 1 and 2 are valid for the multiple Rademachersystem ri(s)rj(t)

i,j=1 considered on the square I I, I = [0, 1], as well. This follows from theequivalence of the symmetric norms for the series with respect to the systems rirj